By also looking at the fermion shifts collected in the appendix, is interesting to notice that all the vacua we found show spectra that do not depend on additional parameters except for a few masses (or the cosmological constant, if different from zero).
This means that for all the gaugings considered the vacua appear in a unique theory with that gauge group and there are no continuous families of models with the same gauge group containing such vacua.
This differs from what was discovered in the 4-dimensional case [1]}, [2]}, where it was found that one can have infinite families of gaugings with the same gauge group and vacua whose existence and whose value of the cosmological constant may depend on the parameter specifying the family of gaugings.
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